Problem number 1.21 only. Thanks!
![CHAP. 1]<br>VECTOR ANALYSIS<br>11<br>1.20. Given the plane 4x + 3y + 2z = 12, find the unit vector normal to the surface in the direction away<br>from the origin.<br>Ans. (4a, + 3n, + 2n,)/V20<br>1.21. Find the relationship which the cartesian components of A and B must satisfy if the vector fields are<br>everywhere parallel.<br>A<br>Ans.<br>В, В, В,<br>1.22. Express the unit vector directed toward the origin from an arbitrary point on the line described<br>by x=0, y=3.<br>-3m, – zm,<br>V9 + z<br>Ans.<br>1.23. Express the unit vector directed toward the point (x,, y, z.) from an arbitrary point in the<br>plane y --5.<br>(x, – x)m, + (y, + 5)a, + (z, - z)m,<br>V(x, – x)' + (y, + 5)' + (z, – z)<br>Ans<br>1.24. Expreas the unit vector dirccted toward the point (O,0, h) from an arbitrary point in the plane --2.<br>-Xn, - ya, + (h + 2)m,<br>V + y' + (h + 2)?<br>Ans.<br>Given A-Sn, and B-4a, + B,,, find B, such that the angle between A and B is 45°. If B also<br>has a term B,](https://s3.us-east-1.amazonaws.com/storage.unifolks.com/qimg-008/008_grysxp0-zhwfu0xp.jpeg)
Extracted text: CHAP. 1] VECTOR ANALYSIS 11 1.20. Given the plane 4x + 3y + 2z = 12, find the unit vector normal to the surface in the direction away from the origin. Ans. (4a, + 3n, + 2n,)/V20 1.21. Find the relationship which the cartesian components of A and B must satisfy if the vector fields are everywhere parallel. A Ans. В, В, В, 1.22. Express the unit vector directed toward the origin from an arbitrary point on the line described by x=0, y=3. -3m, – zm, V9 + z Ans. 1.23. Express the unit vector directed toward the point (x,, y, z.) from an arbitrary point in the plane y --5. (x, – x)m, + (y, + 5)a, + (z, - z)m, V(x, – x)' + (y, + 5)' + (z, – z) Ans 1.24. Expreas the unit vector dirccted toward the point (O,0, h) from an arbitrary point in the plane --2. -Xn, - ya, + (h + 2)m, V + y' + (h + 2)? Ans. Given A-Sn, and B-4a, + B,,, find B, such that the angle between A and B is 45°. If B also has a term B,",, what relationship must exist between B, and B,? 1.25. Ans. B, - ±4, VB; + B; = 4 1.26. Show that the absolute value of A•B×C is the volume of the parallelepiped with edges A, B, and C. (Hint: First show that the base has area (BXC].) 1.27. Given A-2a, -,, B-3m, +a,, and C=-2a, + 6a, - 4a, show that C is 1 to both A and B. 1.28. Given A-, -a,, B= 2n,, and C=-, + 3a,, find A BXC. Examine other variations of this scalar triple product. Ans. -4. t4 1.29. Using the vectors of Problem 1.28 find (AX B) x C. Ans. -8a, 1.30. Find the unit vector directed from (2, –5, –2) toward (14, –5, 3). 5 , + 13 12 Ans. 13 1.31. Find the vector directed from (10, 3x/4, x/6) to (5, x/4, x), where the endpoints are given in spherical coordinates. Ans. -9.66m, – 3.54m, + 10.61a, 1.32. Find the distance between (2, x/6, 0) and (1, A, 2), where the points are given in cylindrical coordinates. Ans. 3.53 1.33. Find the distance between (1, x/4, 0) and (1, 3x/4, x), where the points are given in spherical coordinates. Ans. 2.0 1.34. Use spherical coordinates and integrate to find the area of the region 0spsa on the spherical shell of radius a. What is the result when a= 2x? Ans. 2aa', A = 4ta? 1.35. Use cylindrical coordinates to find the area of the curved surface of a right circular cylinder of radius a and height h. Ans. 2nah